Stress-Strain
Theory
Under action of applied forces, solid bodies
undergo deformation, i.e., they change shape
and volume. The static mechanics of this
deformations forms the theory of elasticity,
and dynamic mechanics forms elastodynamic
theory.
Strain Tensor
After deformation
u(x+dx)
dx
x
dx’
u(x)
dx
dx’
x’
Displacement vector: u(x) = x’- x
2
2
2
2
Length squared: dl = dx1 + dx2 + dx3 = dxi dx i
2
2
dl = dx’
dx’
=
(du
+dx
)
i
i
i
i
= dui dui + dxi dxi + 2 du i dx i
Strain Tensor
After deformation
u(x+dx)
dx
x
dx’
u(x)
dx
dx’
x’
2
2
2
2
Length squared: dl = dx1 + dx2 + dx3 = dxi dx i
2
dl = dx’
i dx’
i = (dui +dx i )
2
= dui dui + dxi dxi + 2 du i dx i
(1)
Length change: dl2 - dl2 = dui dui + 2du i dx i
dui = dui dxj
into equation (1)
Substitute
dx j
Strain Tensor
After deformation
u(x+dx)
dx
x
dx’
u(x)
dx
dx’
x’
Length change: dl2 - dl2 = dui dui + 2du i dx i
dui = dui dxj
into equation (1)
Substitute
dx j
Length change: dl2 - dl2 = Ui U i
=
Strain Tensor
(1)
(2)
(dui + du j + duk du k )dx i dx j
dx j dx i dx i dx j
Problem
V>C
1 light year
Problem
V>
<C
1 light year
Elastic Strain Theory
Elastodynamics
Acoustics
e
xx
dL
L’-L
=
=
=
L
L
L’
L’
L
Length Change
Length
Acoustics
e
xx
dL
L’-L
=
=
=
L
L
L’
L’
L
Length Change
Length
Acoustics
e
xx
dL
L’-L
=
=
=
L
L
L’
L’
No Shear Resistance = No Shear Strength
L
Length Change
Length
Acoustics
dw
dw, du << dx, dz
dz
Tensional
dx
du
Acoustics
really small
big +small big +small
Area Change
Area
=
(dz+dw)(dx+du)-dxdz
dx dz
=
dxdz+dxdw+dzdu-dxdz
+ O(dudw)
dx dz
dw
=
=
dz
dx
du
Infinitrsimal strain
assumption: e<.00001
dw du
+
dz
dx
=
e
zz
+
U
e
xx
Dilitation
1D Hooke’s Law
pressure
strain
F/A = -k dx
du
=
P= -
k
U
( e zz +
Infinitrsimal strain
assumption: e<.00001
e
xx
)
Bulk Modulus
Pressure is F/A of outside
media acting on face of box
Hooke’s Law
Dilation
F/A = k ( e zz +
Bulk Modulus
e
k U
P = -k ( e + e )+ S
xx
)
Infinitrsimal strain
assumption: e<.00001
=
zz
xx
Source or Sink
Larger
k
= Stiffer Rock
Compressional
Newton’s Law
ma = F
..
dP
r u = - dx
..
rw
;
density
Larger
P (x,z,t)
k
=
dP
- dz
..
Net
-dxdz
r force = [P(x,+dx,z,t)-P(x,z,t)]dz
ux,
= Stiffer Rock
P (x+dx,z,t)
Newton’s Law
1st-Order Acoustic Wave Equation
..
u=(u,v,w)
ru = P
..
..
dP
dP
ru = rw = ;
dx
dz
density
Larger
P (x,z,t)
k
= Stiffer Rock
P (x+dx,z,t)
Newton’s Law
1st-Order Acoustic Wave Equation
..
ru = (1)
P (Newton’s Law)
..
(2)
P
=-k
..
U
(Hooke’s Law)
Divide (1) by density and take Divergence:
..u = - 1
[
r
(3)
P]
Take double time deriv. of (2) & substitute (2) into (3)
..
(4)
P = -k
[1
r
P]
Newton’s Law
2nd-Order Acoustic Wave Equation
.. 1
k
P]
[
P =
r
Constant density assumption
..
k
P = r
2
c =
Substitute velocity
..
P
k
r
P = c
2
2
P
Summary
1. Hooke’s Law: P = - k
2. Newton’s Law:
U
..
ru = -
P
3. Acoustic Wave Eqn:
..
P = -k
P]
[1
r
Constant density assumption
2
c =
k;
r
..
P = c
2
2
Body Force Term
P +F
Problems
1. Utah and California movingE-W apart at 1 cm/year.
Calculate strain rate, where distance is 3000 km. Is it exx or exy ?
2. LA. coast andSacremento moving N-S apart at 10 cm/year.
Calculate strain rate, where distance is 2000 km. Is is exx or exy ?
3. A plane wave soln to W.E. is u= cos (kx-wt) i.
Compute divergence. Does the volume change
as a function of time? Draw state of deformation boxes
Along path
Divergence
U
U = Alim0 U n dl
A
+ U(x,z+dz)cos(90)dx
>>
= 00
dxdz
- U(x,z)dz
dxdz
n
= U(x+dx,z)dz + U(x,z+dz)cos(90)dx
dxdz
dxdz
U(x,z)
P = -k (e
U(x+dx,z)
(x+dx,z+dz)
n
(x,z)
e
No sources/sinks
inside
box.
Sources/sinks
inside
box.
+
zz come
Whatgoes
goesin
inmight
must
outxxout
What
not come
)